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Model a torch battery of length l to be made up of a thin cylindrical bar of radius ‘a’ and a concentric thin cylindrical shell of radius ‘b’ filled in between with an electrolyte of resistivity ρ (see figure). If the battery is connected to a resistance of value R, the maximum Joule heating in R will take place for :

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a

R=ρ2πlba

b

R=ρ2πllnba

c

R=2ρπllnba

d

R=ρπllnba

answer is B.

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Detailed Solution

Maximum joule heating in R is possible when internal resistance = external resistanceConsider a vertical cylindrical strip of thickness dx and radius x.Its small resistance dR can be given as, dR=ρdx2πxlAfter integrating we get, Rinternal=∫ρabdx2πxl=ρ2πl abdxx=ρ2πllnxab=ρ2πllnb−lna=ρ2πllnba=Rexternal

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Model a torch battery of length l to be made up of a thin cylindrical bar of radius ‘a’ and a concentric thin cylindrical shell of radius ‘b’ filled in between with an electrolyte of resistivity ρ (see figure). If the battery is connected to a resistance of value R, the maximum Joule heating in R will take place for :